Título inglés |
Geodesic flow on *SO(4)*, Kac-Moody Lie algebra and singularities in the complex t-plane. |

Título español |
Flujo geodésico en *SO(4)*, álgebra de Lie de Kac-Moody y singularidades en el t-plano complejo. |

Autor/es |
Lesfari, Ahmed |

Organización |
Dép. Math. Fac. Sci. Univ. Chouaib Doukkali, El Jadida, Marruecos |

Revista |
0214-1493 |

Publicación |
1999, 43 (1): 261-279, 24 Ref. |

Tipo de documento |
articulo |

Idioma |
Inglés |

Resumen inglés |
The article studies geometrically the Euler-Arnold equations associatedto
geodesic flow on *SO(4)* for a left invariant diagonal metric. Such metric were first introduced by Manakov [17] and
extensively studied by Mishchenko-Fomenko [18] and Dikii [6]. An essential contribution into the integrability of this problem was also made by Adler-van Moerbeke [4] and Haine [8]. In this problem there are four invariants of the motion defining in C^{4} = Lie(*SO(4)* Ä C) an affine Abelian surface as complete intersection of four quadrics. The first section is devoted to a Lie algebra theoretical approach, based on the Kostant-Kirillov coadjoint action. This method allows us to linearize the problem
on a two-dimensional Prym variety Prym_{σ}(C) of a genus 3 Riemann surface C. In section 2, the method consists of requiring that the general solutions have the Painlevé property, i.e., have no movable singularities other than poles. It was first adopted by Kowalewski [10] and has developed and used more systematically
[3], [4], [8], [13]. From the asymptotic analysis of the differential equations, we show that the linearization of the Euler-
Arnold equations occurs on a Prym variety Prym_{σ}(Γ) of an another genus 3 Riemann surface Γ. In the last section the Riemann surfaces are compared explicitly. |

Clasificación UNESCO |
120109 ; 120404 |

Palabras clave español |
Métricas riemannianas ; Variedad riemanniana ; Algebra de Lie ; Flujos geodésicos |

Código MathReviews |
MR1697525 |

**Acceso al artículo completo** |